Abstract.
When
simulating musical instruments it is often
necessary to adjust the tone colour or timbre of existing
sound samples, or to produce entirely new ones. Conventionally, this
requires the creation of new harmonic spectra to
represent the new tone colour, and the modified samples are then generated using additive synthesis. However,
creating the desired spectra is time consuming and laborious especially when they
comprise many
harmonics, and it also requires a lot of skill and experience. Similar
parameter-overload problems apply to physical modelling synthesis. So some means is
desirable to reduce the labour involved, and the
technique of Trendline Synthesis described in this article offers this
advantage. It enables a wide range of prototype spectra to be
created instantaneously by specifying no more than four parameters for each
one, regardless of how many harmonics they might contain.
Thus manual intervention is minimal, making the design cycle for a new synthetic
sample set much faster, cheaper and more flexible than creating it from recordings of existing
instruments or constructing a set of new spectra harmonic by harmonic. Audio
recordings are included, showing that Trendline Synthesis can
produce convincing aural examples of the four classes of pipe organ tone colour - diapasons
(or principals), strings, flutes and reeds. These advantages ensue from the simple means by which a spectral
envelope is approximated by a set of trendlines, and for organ pipe spectra it has been found that only two
lines are necessary.
Because only two parameters are required to define a straight line, it follows
that no more than four are needed to define both trendlines in any spectrum and thus
the spectrum itself. A novel computer design tool is described which facilitates
the process.

When
simulating musical instruments digitally, instances often arise when it is
necessary to rapidly adjust the tone colour of existing
sound samples, a process known as voicing in the organ world. Or a
simulation might need to be developed from scratch for which no sound samples are
available. This can occur when entirely new and 'invented' sounds are
required, and in
such cases sample sets from an existing instrument cannot be created by
definition. Even when samples could be obtained in principle from
recordings of a real acoustic instrument, the process of producing a high
quality sample set is expensive, labour intensive and the result is inherently
inflexible because it is difficult to make other than minor changes.
In circumstances such as these it is frequently necessary to create new harmonic spectra to
represent the required tone colours, or to modify existing spectra by adjusting
their harmonic amplitudes. This too requires a lot of toil. These operations have to be done in the
frequency domain because
it is impossible to modify the time domain
waveforms used in sampled sound synthesis directly. Therefore the new or modified
samples have to be generated using additive synthesis operating on the new or modified harmonic
spectra. In a real time additive synthesis instrument, this step would
take place within the instrument itself.

There
is nothing wrong in theory with this approach, and it is a conventional and
widely used procedure. However in practice,
creating the desired harmonic spectra is time consuming especially when they
comprise many
harmonics, and it requires a lot of skill and experience. Similar problems
apply to physical modelling synthesis, in which a large number of parameters are
invariably required to define a particular tone colour. For instance, the
Viscount 'Physis' system uses 58 parameters for each simulated pipe tone [5]. So some means is
desirable to reduce the labour involved, and the
technique of Trendline Synthesis described in this article offers this
advantage. It enables a wide range of prototype spectra to be
created instantaneously by specifying no more than four parameters for each
one, regardless of how many harmonics they might contain. The
harmonic amplitudes in the spectra are then converted, again instantaneously and automatically,
to time domain wave (PCM) files for exporting to a sound sampler. Loop points
can also be computed automatically and embedded in the wave files if desired.
Thus manual intervention is minimal, making the design cycle for a new sample
set much faster, cheaper and more flexible than creating it from recordings of existing
instruments. This article shows how Trendline Synthesis can generate
convincing examples of the four classes of pipe organ tone colour - diapasons
(or principals), strings, flutes and reeds - each of which only requires four parameters to define the tone colour of
the sample. Some audio recordings are included.

All
the advantages outlined above follow from the simple means by which a spectral
envelope is approximated by a set of trendlines. Any number of lines can be used,
but for organ pipe spectra it has been found that only two are necessary.
Because only two parameters are required to define a straight line, it follows
that no more than four parameters are needed to define both lines in any spectrum, hence
the economy and simplicity of the method. It has been found that
approximating to a set of harmonic amplitudes in a spectrum by using trendlines
results in little change to the tone colour. Such changes as do occur are
eclipsed in any case by those which arise naturally between one note and
the next in real musical instruments such as the pipe organ.

The
novelty of the method relates to simulating the pipe organ, where the use of
trendlines to approximate to the spectral envelopes of organ pipes does not
appear to have been used previously. Other important aspects include the
mandatory representation of power spectra using log-log spectral plots rather
than either or both axes (power and frequency) employing linear scales.
Only with a log-log plot does the simplicity of the technique emerge in which
only two linear trendlines, rather than a multiplicity, are necessary.

An
earlier article on this site introduced the concept of trendlines to approximate to
the spectral envelopes of groups of harmonics in an organ pipe spectrum [1],
though it can also be applied to other musical.instruments. The general idea is
illustrated at Figure 1.

This
diagram shows the harmonic amplitudes of an organ Trumpet pipe by Rushworth
& Dreaper (blue dots), together with trendlines applied to two
easily-identifiable groups of harmonics. The first group includes the low
order harmonics having amplitudes comparable with the fundamental, whereas the
second group comprises those of higher order which exhibit rapidly diminishing
amplitudes. Each trendline was fitted to its group of harmonics using a
least-squares procedure. The knee or breakpoint between the two lines,
denoting that frequency at which the second takes over from the first, lies near
the fifth and sixth harmonics in this case. The previous article [1]
demonstrated that spectral groups and trendlines could be identified for each of
the four classes of organ tone - diapasons (or principals), flutes, strings and
reeds. It also put forward the view that convincing aural reconstructions
of the pipe sounds could be achieved using the trendline parameters alone,
rather than the usual technique of generating a sound sample by applying
additive synthesis to all of the actual harmonics. This view was arrived
at after many listening tests, and it represents a considerable simplification
when a spectrum contains many harmonics. It was found that
approximating to a set of harmonic amplitudes in a spectrum by using trendlines
results in little change to the tone colour. Such changes as do occur are
eclipsed in any case by those which arise naturally between one note and
the next in real musical instruments such as the pipe organ.

This
article takes the analysis further into the realm of digital musical instruments.
Besides the economies just mentioned, the use of spectral envelopes in the form
of trendlines enables a wide range
of musical instruments, not just pipe organs, to be simulated
readily. Moreover, rapid appraisal and readjustment of the tone colours or
timbres becomes
possible without having to painstakingly amend some or all of the individual
harmonics in a spectrum, even if one knew which ones were relevant. In the
particular case of synthesising pipe organ tones digitally, this ability to
rapidly 'revoice' an instrument or to create a new one from scratch does not
require the expensive, labour intensive and inherently inflexible process of
constructing a fixed sample set from recordings of individual organ pipes. This
is also an advantage when a new simulation is to be developed for which samples
are unavailable, whether of a pipe organ or any other musical instrument.
Entirely new and 'invented' sounds can also be generated very easily, and in
such cases sample sets from an existing instrument cannot be created by
definition. Thus Trendline Synthesis is presented here as a new and
flexible music
synthesis technique with wide potential application. Although additive
synthesis was mentioned above, Trendline Synthesis is not additive
synthesis. It merely uses additive synthesis as part of a new process of
generating waveform samples
which can then be used in a sound sampler. Nevertheless, because a set of
harmonic amplitudes is generated as part of the Trendline Synthesis process,
these can also be used in a real time additive synthesis sound engine should this be
desired.

All
the advantages outlined above follow from the simple means by which a spectral
envelope is approximated by a set of trendlines. Any number of lines can be used,
but for organ pipe spectra it has been found that only two are necessary.
A straight line is fully defined using only two parameters
- its gradient or slope and its intercept on the vertical axis. This is
enshrined in the standard equation of a straight line, written as y =
mx + c where m is the gradient and c the intercept (here the
vertical axis is taken as the y-direction and the horizontal axis as the x-direction).
In practice other convenient parameters can also be used provided they reduce to
the form just outlined with two independent (unrelated) values for each
line. Thus in this article, a trendline is defined by its slope (specified
in decibels per octave - dB/8ve) and the breakpoint or harmonic number at which
it ends (for the Group1 line) or at which it starts (for the Group 2
line). The breakpoint is not restricted to lie at an integer harmonic
value as it can assume any intermediate point between a pair of adjacent
harmonics. Thus
two intersecting trendlines can be fully defined on a spectrum plot using just three
numerical parameters - their slopes and the common breakpoint (the point of
intersection) between them.
Using these lines, two sets of harmonic amplitudes can then be generated, each set lying on
one of the lines. An example will now be discussed.

Figure
2. A typical trendline plot

Figure
2 shows a spectrum with two trendlines. The first line has an upwards
(positive) slope of 3 dB/8ve and the second has a downwards (negative) slope of
-17 dB/8ve. The breakpoint is at a 'harmonic number' of 3.5. The
quote marks serve to show that this is, in a sense, a fictitious number because it does not correspond to an
actual harmonic, lying as it does between harmonics three and four, but it is
more flexible for the breakpoint not to be restricted to integer values.
There are 42 harmonics enclosed by the two trendlines. This type of graph is the most important
output screen of an interactive design program which I
have developed (written in C). The diagram does not correspond to any particular
organ tone as it is shown here purely for illustrative purposes, though it relates
more closely to the harmonic recipe of a string-toned pipe rather than to any
other type. These almost invariably have the upwards slope as shown for
the first few (Group 1) harmonics, followed by a steeper descending slope
associated with the remaining (Group 2) harmonics. The total number of
harmonics, 42, is also typical of pipes with a keen string timbre. Note that the number of
harmonics is not required of the user, as this parameter drops out of the process
automatically as a consequence of merely having defined the two trendlines.
Thus if you want a keener or brighter tone, you simply adjust the lines to give
you more harmonics in one region of the spectrum or the other, or both.

So
what happens next? What does one do with this picture? It has
already been stated that it is only necessary for the user to
specify the slopes of the two lines and their breakpoint, that is, to input just
three numbers. The screen then appears and harmonics
are drawn in automatically, as shown by the red lines in the diagram, and
their amplitude values are saved to a file in case one wants to call up a
particular spectrum again or for other purposes. These values are also used to
generate a Windows wave file by additive synthesis (a standard PCM file with a WAV extension)
which can then be auditioned in a sound sampler or wave editor. For this to happen, loop points are
created automatically so that the sound lasts for as long as the user
wishes. The same looped WAV file can also be imported into one of many
external rendering engines such as a software synthesiser or an existing
digital organ system, when other articulation parameters such as attack and
release envelopes will be added to complete the voicing of the sample.

Note
the use of a log-log spectrum plot in which the quantities on both axes are
represented logarithmically. This is essential, otherwise the best-fit spectral
envelopes turn out to be curves rather than straight lines and more parameters
would therefore be required to specify them as (for example) polynomials.
Although not directly relevant to this article, it is of interest that the ear
and brain process both the amplitude and frequency of sounds
logarithmically. It is also not without interest that the simple straight
line representation of spectra at the heart of Trendline Synthesis is capable of
producing results which satisfy the ear. Therefore there might be
implications here for the neural mechanisms of musical perception which were
drawn out in the earlier article [1].

Although
the process of designing sounds using trendlines is essentially simple provided
one has the computer-based tools available, it will probably be difficult for a
novice to get the sounds s/he wants at first. Some practice is required,
and in this respect the technique is no different to any other form of musical
instrument synthesis, and indeed to the art of voicing real organ pipes.
It is also helpful, if not essential, to have as much experience as possible of
what organ pipe spectra look like. This being so, we
now look at some specific examples of trendline plots for each of the four classes of
organ tone - diapasons (or principals), strings, flutes and reeds - to give a
flavour of what the design process entails.

Figure
3 is a plot which generates an Open Diapason tone. It relates to
a pipe in the middle of the organ key compass, around middle F sharp, and this
is true of all the examples to be discussed. This is an important point
since the spectrum of organ pipes varies systematically across the compass as a
result of the varying scales of the pipes (scale is a measure of cross-sectional
area to length). Thus it is usually not possible to use the same spectrum
or harmonic recipe, and thus the same trendline parameters, across the entire
compass. Scaling is discussed in more detail later in the article The parameters
of the plot were chosen to produce the somewhat subdued, dignified diapason tone
typical of a British organ of the first half of the twentieth century rather
than that of the brighter and more zestful principals of a German or Dutch
instrument.

Figure
3. Trendline plot for an Open Diapason

There
are 13 harmonics, all of which lie on the same descending line. This
picture was generated using the following parameter set:

Breakpoint
at harmonic number: 1

Slope
of Group 1 trendline: -16 dB/8ve

Slope
of Group 2 trendline: -16 dB/8ve

The
use of a breakpoint at the first harmonic effectively collapses the Group 1
spectral region and its trendline to nothing, but it is still necessary to
specify its gradient to satisfy the syntax of the design program. Thus
in practice we are only using a single trendline here to generate the sound of
this particular Open Diapason, which demonstrates the economy and efficiency of
Trendline Synthesis as a means of designing musical sounds.

Figure
5 shows a trendline plot for a stopped flute pipe of the genre often called
a Stopped Diapason. This is an unfortunate misnomer as the tone has
nothing of the qualities of any other sort of diapason because the pipe has a
flute-like tone. The presence of
the stopper causes the even-numbered harmonics of the real organ pipe to be
suppressed relative to the odds, giving the pipe a characteristically hollow,
'woody' sound for reasons explained in reference [2]. In
the design program used here, even harmonic suppression is achieved by
introducing a fourth parameter. As before, the spectrum corresponds to the middle
region of the key compass.

Figure
5. Trendline plot for a Stopped Diapason

There
are 9 harmonics, some of which have negligible amplitudes. As with the
Open Diapason, only one trendline is used in effect, this being achieved by
specifying a breakpoint at the first harmonic in the same way as before. This
picture was generated using the following parameter set:

Breakpoint
at harmonic number: 1

Slope
of Group 1 trendline: -18 dB/8ve

Slope
of Group 2 trendline: -18 dB/8ve

Even
harmonic suppression: 20 dB

Note
the extra parameter introduced here which suppresses the even-numbered harmonics
relative to the odds by the specified amount. (When using the trendline design program this
parameter always has to be given a value, and in the previous cases it was
merely set to zero).

Two
reed tones are presented here, a Trumpet and a Clarinet. Note these are
designed as versions of these sounds familiar to those in the organ world rather
than representing the eponymous brass and woodwind instruments. Although
there are some passing similarities, the two types of tone (organ and
orchestral) are more often characterised by their differences.

Figure
6 shows a trendline plot for a Trumpet pipe having a fairly bright, assertive
and brassy tone. In an organ it might attract a name such as Fanfare
Trumpet. As before, the spectrum corresponds to the middle
region of the key compass.

Figure
6. Trendline plot for a Trumpet

There
are 37 harmonics, distributed within the regions enclosed by two trendlines. This
picture was generated using the following parameter set:

Breakpoint
at harmonic number: 7

Slope
of Group 1 trendline: -3 dB/8ve

Slope
of Group 2 trendline: -22 dB/8ve

Even
harmonic suppression: 0 dB

The
'even harmonic suppression' parameter, introduced for the Stopped Diapason
above, is again included though it had no effect in this case as it was set to zero.

Figure
7 shows a trendline plot for a Clarinet pipe. It probably inclines more
towards the somewhat thin and penetrating tone of a Corno di Bassetto or even a
Krumhorn rather than the smoother, more reticent type of Clarinet often
encountered. As before, the spectrum corresponds to the middle
region of the key compass.

Figure
7. Trendline plot for a Clarinet

There
are 23 harmonics, distributed within the regions enclosed by two trendlines. This
picture was generated using the following parameter set:

Breakpoint
at harmonic number: 7

Slope
of Group 1 trendline: -3 dB/8ve

Slope
of Group 2 trendline: -30 dB/8ve

Even
harmonic suppression: 15 dB

As
with the Stopped Diapason, the 'even harmonic suppression' parameter plays an
important role here. Note how qualitatively similar the trendline
structure is to that of the Trumpet, the only numerical difference being the
somewhat more rapid fall-off for the Group 2 harmonics. But it is mainly
the introduction of even harmonic suppression which makes the Clarinet sound so
utterly different to the Trumpet.

At
this point it might be appropriate to listen to what these examples actually
sounded like when the synthesised samples were imported into a sound
sampler. The processes required to generate them were described
above, but to recapitulate the following steps were involved:

1.
Design the trendline spectra on-screen as just described.

2.
Using the harmonic amplitudes thus defined, generate samples (wave files) at the desired
frequency (defined by MIDI Note Number) for each tone
using additive synthesis. Having input the Note Number this is done automatically by the design program
- there is no messing about trying to read off the harmonic amplitude values using a cursor, so it
is an almost instantaneous process. The harmonic amplitudes are also saved
in a separate file for later use if required.

3.
Add loop points and embed them in the sample files (the loop points are also
identified automatically by the
design program).

4.
Export the samples to an external sound sampler.

The
following mp3 file contains the five examples discussed above and in the same
order - Open Diapason, Viol d'Orchestre, Stopped Diapason, Trumpet and
Clarinet. Multiple but slightly different instances of each of these tones were
generated to simulate the scaling of the corresponding organ pipes across the
keyboard, as discussed in more detail below. Additionally, voicing parameters were applied in the sampler
itself to simulate proper attack and release envelopes, volume levels and the other necessary
features of real organ pipe sounds. Thus each tone is represented as a complete organ stop in the sampler, and the hymn
tune 'Moscow' is played on each one in turn. The two reed examples are
played as solos accompanied by one of the flue stops so that the tones can be
better appreciated.

I
find it remarkable that such a wide range of tones can be obtained merely by
varying the four parameters of so simple a model, and to my mind it might be
saying something about how the ear and brain perceive musical sounds. This
idea was developed further in the earlier article [1].

We
now come to some additional issues which go beyond synthesising the samples themselves.
None of them are specific to Trendline Synthesis because
they are common to any other method of generating synthetic samples, so they will not be discussed in
excessive detail. Scaling is one of these
topics. This is a complex subject and further
information can be found elsewhere on this website, for example at reference [2]
(see the section entitled Pipe Scales). Briefly, the scale of a
cylindrical organ pipe is the ratio of its diameter to its length.
Constant scale means that the ratio remains the same across all the pipes
which constitute an organ stop, but this is never used. If it were used, the pipes
would get too narrow towards the middle and treble end of the key compass.
In this case, not only would the overly-narrow pipes eventually cease to emit enough acoustic power but they would sound thin and shrill on account of
the excess of harmonics they radiate. This would happen because the number
of harmonics emitted by an organ pipe depends on its cross-sectional area, with
narrower pipes emitting more harmonics and vice versa. Therefore a variety
of non-constant scaling laws is used in which the variation of pipe diameter does not follow the
same mathematical progression which governs pipe length. Ascending the key compass
from the bass end, pipe length halves every octave (every twelfth note) but pipe
diameter reduces
more slowly. Typically, the diameter halves every sixteenth note or so
for cylindrical flue pipes, with the result that they remain progressively wider as they ascend the compass than they would
if constant scaling were used. Thus the effect of scaling is to reduce the
number of harmonics which the treble pipes emit and to simultaneously increase the
acoustic power which they radiate. Conversely, the bass pipes are narrower
(relative to the middle of the key compass) than they would be if constant scaling were used, meaning that they have less of
a tendency to overwhelm those higher up the keyboard. They also emit
somewhat more harmonics than they otherwise would.

To
create a complete synthetic organ stop, multiple samples have to be generated
so that they can be scaled across the key compass just as the pipes themselves
are scaled in a pipe organ. Desirably, a separate sample should be
provided for each note rather than 'stretching' a single sample across a group
of adjacent notes, or interpolating the waveforms between two
adjacent samples which might be spaced widely across the key compass. Both
of these 'short cut' techniques are used often in commercial digital musical
instruments. Therefore
the spectra for the multiple samples generated for a given simulated organ
stop also have to be scaled or 'shaded' to produce the same subtle variation of
tone quality across the keyboard as pipe scaling does in a pipe
organ. In other words, the same spectrum
cannot be used to generate a complete set of samples for realistic simulation of
a given organ stop. Because more harmonics
are required towards the bass, this has to be reflected in the spectra and hence
in the trendline parameters for the corresponding sample files. Towards the
treble the reverse applies. Achieving this is a matter of
considerable subtlety and it requires some practice and experience to get it
right. A good indication of what is required can be found by examining real
organ pipe
spectra at various points across the keyboard, as it is not something which can be captured in a simple equation or
for which simple instructions can be given. To use an appropriate pun,
scaling is an example of 'sound design', and it separates the better digital
musical instruments (and their tonal designers) from the rest.

How many samples are required for each simulated
stop? As mentioned above, ideally there should be one sample per note, just as a pipe organ
has one pipe per note, with the trendline parameters for each sample being
shaded appropriately across the compass. This is no different to creating
a sample set based on recordings of real pipes, in which there should ideally be
an independent sample for each note of each stop. However, applying the
individual shadings to the trendline parameters is a very much quicker process
than laboriously recording, denoising and otherwise preparing a sample set made
from acoustic recordings of the pipes.

A
more recent article on this site discusses the scaling of synthetic samples in
detail [4].

Any synthetic sample
when first generated has none of the random variations which can sometimes be
heard in real
organ pipe sounds, whether it be generated using Trendline Synthesis or any
other method. Such variations include small changes in tone quality
from note to note, and slight real-time perturbations in amplitude and frequency
while a note is sustained. However
in my opinion these, and the subject
itself, often take on an unreal and bizarre Emperor's New Clothes quality in
some quarters. Sometimes an exaggerated posture seems to be adopted that,
because a particular organ system is capable of rendering the variations, then
you are darn well going to have to listen to them! (Much the same attitude
surfaces regarding equally exaggerated attack and release transients). So instead of allowing the issue to develop a conflated life of its own
it is better to focus on reality. As an example, recently I heard an
organ built in 1858 and meticulously restored by a master organ builder (Mander), in which there was absolutely no
random variation detectable in the notes comprising an Oboe solo. The same
sounds could quite easily have been generated by the simplest
digital sample looped over just a single cycle of the waveform, and it would
have been impossible to tell the difference! Therefore it seems to me that
certain sections of the digital organ community exaggerate the real
importance of randomness, perhaps because so many sample sets recorded from organs
are in fact of poor quality because the pipe organ itself was not in good
condition. There is no doubt that some sample sets are unsatisfactory,
and of course random variations in such cases will often be pronounced and impossible
to ignore. But this is unacceptable, particularly in those cases where one has
to pay for them. Minutely simulating an organ
whose pipes, action and winding system are in poor condition is perverse. Life is too short to waste time, effort and money in this way.
Even if randomness can be heard on a pipe organ, it is frequently only at short range. At realistic listening distances and in
reasonably reverberant conditions the variations frequently cannot be
discerned. Similarly, the majority of organ music does not consist of
homophonic single-note solos of long duration - it is polyphonic and moves
faster, and it then becomes impossible to detect the random variations of each
note even if they are present.

Nevertheless
there is some benefit to be gained from injecting a sensible measure of
randomness into a sample set which is generated purely synthetically. One simple way to do this is to
vary the deterministic and strongly-correlated harmonic amplitudes initially
resulting from Trendline Synthesis from note to note across the keyboard.
Randomising the amplitudes of individual harmonics within a certain range, say
up to ±
5 dB
of the values initially calculated, can be done automatically and very easily. The resulting spectra then have some of the amplitude scatter which real pipe spectra exhibit
from note to note. However it is seldom necessary to do this because
scatter will be imposed naturally on the spectra of all the samples when they
are replayed using loudspeakers in any room, owing to the reflections occurring
at the room boundaries. Only when listening using headphones will this not
happen. Another technique is to modulate the instantaneous amplitude and frequency of
each sample slightly using a random number sequence while the sample is
sounding. This can either be done in real time by the sampler if it is
capable enough, or the variations can be impressed off-line on the samples
before they are imported into the sampler. The sampler can also be
programmed to apply one-off small changes to the amplitude and frequency of each
sample - note that this is additional to the real time variations just
mentioned. The frequency variations reflect the fact that no real musical
instrument will ever be perfectly in tune within itself. Yet another option is simply to add wind noise.

In
addition, audible attack and release transients can be added, though not to all samples
otherwise the effect is identifiably unreal. An example of the way I
synthesise attack transients is described elsewhere on this site in reference [3]
(see the section entitled Attack Transients).

This
article has demonstrated a wide range of authentic organ pipe sounds which can
be simulated using Trendline Synthesis, embracing any conceivable timbre in
addition to the examples described here. Although the underlying
concepts are straightforward, the necessity for custom software presents a
barrier to those wishing to evaluate the technique for themselves.
Consequently a free 'starter toolkit' has been put together whose main component
is a program which enables a spectrum to be generated once the four parameters
described above have been defined. The spectrum is displayed on the
monitor just as in the examples shown earlier, and it can optionally be saved
for later use. Also as described earlier, the corresponding sound sample
in the form of a Windows WAV file can be generated and saved. The WAV file
contains embedded loop start and end points which are calculated automatically,
thus the sample can be auditioned immediately in a wide range of sound samplers
and wave editors without requiring the user to fiddle about searching for loop
points. In addition to this program, several utility applications are also
included for convenience. The software should run on Windows computers
from XP onwards.

The
software is licensed for personal and non-commercial use only and you are
requested to read the End User License Agreement contained in the package.
Note also that no responsibility is accepted for any unwanted consequences
arising from the use of this software. You should also ensure that the
download is virus-free before unzipping it.

The
package can be downloaded as a zipped file from the link below. Having
opened it, the file named Trendline-Software-Distro1.pdf should be read
as the instruction manual for the package.

Trendline
Synthesis is an example of spectral envelope processing in which the envelopes
are of the simplest form possible - straight lines. Formally, the envelope
is represented in piecewise linear fashion. It has been found that only two trendlines are
required to characterise the harmonic amplitude envelopes of most, if not all, organ pipe spectra
in the four categories of diapasons (or principals), strings, flutes and
reeds. This means that no more than four parameters are required to define any
organ tone synthetically, regardless of how many harmonics it contains, because a straight
line is specified by two numbers. The harmonic amplitudes in the synthetic spectrum lie on the trendlines, therefore
it is trivially easy to compute a complete synthetic spectrum, and
this can be transformed into the time domain by applying additive synthesis (the
real discrete inverse Fourier transform). It has been found that a very wide range of
prototype organ tone colours can
be generated in this way, that is by varying only a few parameters rather than
attempting to create a complete spectrum each time by laboriously specifying the amplitude of each
harmonic, or by attempting to modify the large number of parameters involved in
physical modelling. A computer design tool has been developed to test these ideas
which generates wave samples for importing directly into a sound sampler.

It
is only possible to use a piecewise linear approximation to the spectral
envelopes of organ pipe sounds when the spectra are mapped into a log-log domain having logarithmic axes for both amplitude and frequency.
Otherwise the trendlines become curved, requiring higher-order polynomials or
other nonlinear functions to represent the approximated envelopes. It is
not known why this happy simplification occurs, but it might be related to the
fact that the ear and brain also process amplitude and frequency
logarithmically. This might have evolved in response to natural
music-like sounds arising in the environment, such as within the vocal
tracts of animals and humans. Independently of such speculation however,
this article has shown unequivocally that the simple straight
line representation of spectra at the heart of Trendline Synthesis is capable of
producing results which satisfy the ear, therefore there might nevertheless be
implications here for the neural mechanisms of musical perception.
Perception involves pattern recognition, the process of identifying incoming
audio information by assigning it to one of a variety of classes such as the
type of musical instrument or organ pipe which generated it. Machine
recognition in artificial intelligence often employs piecewise linear feature
extraction or classification (such as the nearest neighbour classifier), and
there is no a priori reason in principle why brains should not employ
similar mechanisms.

This
paper began by saying that "one of the most challenging tasks in
physically-informed sound synthesis is the estimation of model parameters to
produce a desired timbre". It disclosed that Viscount's physically-modelled
organs use 58 macro-parameters for each simulated pipe, "some of which
are intertwined in a non linear fashion and are acoustic-wise non-orthogonal
(i.e. jointly affect some acoustic feature of the resulting tone)".
That the paper appeared a decade after the introduction of these instruments
demonstrated not only the difficulty of voicing them but the importance of
solving the parameter-overload problem.